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This, too, is a subject carried over from an earlier thread. I could PM Kees (aka DoelKees) about this, but I thought others might be interested to follow the discussion and chime in.

Kees: You mentioned a TuneLab plug-in that you wrote, possibly solving the initial stretch choices in extending the temperament into the rest of the mid-range? As has been discussed here earlier, I think none of us has been able to get the same ETD results as we can get aurally.

As I've followed your posts carefully since you joined here, I know that if you're on to something regarding the maths behind it, it's worth checking out. I don't have TuneLab myself, but I could run it in trial mode on my Pocket PC. Would you like me to measure the tuning result of your plug-in against what I get aurally?

I PM'd you long time ago about getting me a recording of your aural EBVT3 tuning, and am still interested. Strip mute the piano and record each note separated by a silence, and I can take it from there.

I remember trying to reconcile the computed tuning, following Bill's verbal recipe to the letter, with the offsets he provides, using inharmonicity data of the piano brand he tuned, though not the specific piano, and found there was quite a bit of discrepancy. I tried to tweak the math to account for vagueness in Bill's recipe to improve the match. For example what is a "pure 5th". For me it's an equal beating 3:2 6:4 compromise. That seemed to improve things. On high inharmonicity piano's there seems to be no way to tune D5 in a way consistent with Bill's requirements though.

Quite a few people have emailed me their tunelab inharmonicity measurements and I sent them my computed EBVT3 offsets. Responses were very positive.

If you're interested, I can compile a list of ambiguities in your aural EBVT3 recipe which you can perhaps clarify. They are ambiguous from a mathematical point of view, as you sometimes refer to intervals being "improved", "acceptable" etc. Not so easy to program on a stupid computer

I tuned a Yamaha C5 in EBVT III for a performance tomorrow, and recorded all the pitches (x 3) with the muting strips still in.

I also did some 10ths, 12ths and other runs. Too late to put it up on the web tonight, but I might find some time tomorrow.

I did my best to be faithful to the temperament and to be fair in the stretch, and I have to say that the end result was much better than the slightly "watered out" version I've been using lately. I really really like that F-C pure 5th, will go back to the original.

Bill, you truly have a phenomenal temperament! EBVT III, to me, is sheer beauty. If somebody might be able to implement this on an ETD, I bet it's Kees

If you're interested, I can compile a list of ambiguities in your aural EBVT3 recipe which you can perhaps clarify. They are ambiguous from a mathematical point of view, as you sometimes refer to intervals being "improved", "acceptable" etc. Not so easy to program on a stupid computer

Let me know.

Kees

Kees, why don't you take that list into this thread? I'm sure some of us tuning the temperament could contribute to the task of bridging the gap between numbers and adjectives!

Ok Pat, Bill, let's start with step 1 of EBVT. The instructions I have state to tune F3A3 at 6bps instead of the usual 7bps.

Now on all piano's that I have data on the F3A3 third, when tuned in ET, beat from 6.3 bps (on the lowest inharmonicity concert grands) to 6 bps on large uprights to even slower.

So the first adjustment I made is to first compute a standard ET F3A3 for the particular paino, then scale its beat rate for EBVT by a factor 6/7. Usually this comes to be about 5 to 5.3 bps instead of the 6.

This improves dramatically the agreement with my computed tuning chart and the one Bill provided me with for a Steinway D, for which I have two measured instruments.

Needless to say when I check my calculations on a hypothetical piano with no inharmoniciy Bill's number are right on.

I'll post comments on the instruction on how to extend the temperament in a bit. They are crystal clear for aural tuning but computers do not know what a "reasonable" octave is

If somebody might be able to implement this on an ETD, I bet it's Kees

Well, thank you for that vote of confidence.

As far as I know all ETDs use the method of having 12 UT offsets to be added to the offsets related to stretching and inharmonicity, irrespective of the octave. A paradigm shift is needed here as EBVT needs 88 unique offsets, which depend on the inharmonicity.

Currently the only way to implement EBVT with an ETD is in tunelab and it involves emailing the IH data to me, and I compute the custom tuning.

I learned this from Bill in a kind of upside down way: my first EBVT III:s were on grands 6ft+, and 6bps worked fine. 6bps works on shorter instruments, too, but then some of the fifths are heavily compromised (Bill, I don't know if you remember me asking about exactly this in the beginning when I hit a brick wall with a spinet kind of piano). Anyways, I can confirm first hand that Bill subconsciously drops his 6 bps down in speed when he deals with shorter instruments.

The reason I know is because it was the first instrument I ever played that Bill had tuned... At the PTG convention this summer, I played a short Petrov, if my memory doesn't fail me? Anyways, the "balanced frame-work" (F3-A3, C4-E4+, G3-E4+, G3-B3) was there, only slightly slower than 6bps - I heard it as "between 5 and 6 bps, which corresponds well with you math.

This is fascinating, because of course Bill does this without thinking twice about it. Hence the need of an interpreter

Kees, I have on paper, the 2 EBVT III tunings Bill did for me...This later one has much more of a stretch than the first one. These are exact figures for all 88 notes. Of course this was designed for my piano. Would these figures help you in any way?

If somebody might be able to implement this on an ETD, I bet it's Kees

Well, thank you for that vote of confidence.

As far as I know all ETDs use the method of having 12 UT offsets to be added to the offsets related to stretching and inharmonicity, irrespective of the octave. A paradigm shift is needed here as EBVT needs 88 unique offsets, which depend on the inharmonicity.

Currently the only way to implement EBVT with an ETD is in tunelab and it involves emailing the IH data to me, and I compute the custom tuning.

Kees

See, I have this feeling that once we sort all of these adjectives vs. math issues out, your unique calculation could easily be implemented into any of the leading ETD:s out there, if anybody dared. My first hunch was that TuneLab might show the greatest interest (you and many other PW members endorsing it), but that was not the feeling I got from my initial feeler.

Unlike with ET where the contiguous thirds "find" the exact speed of F3-A3, I always set it at 6 beats per second (or at least I perceive it that way when tuning either version of the EBVT. After all, who could tell if it is exactly 6.000 beats per second or not.

The point is that on short scaled pianos, when I use the contiguous thirds, I can hear that the result is a slower beat than the usually perceived 7 beats per second. It is a mistake to impose that beat speed on such a piano. It will skew everything else if that is done. I know that people often do it but that is why they can't get other intervals to work out as they expect afterward. The reason, of course is that on such pianos, the F3 string will often have very high inharmonicity unless it happens to be a wound string.

This may be the reason that some have said that contiguous thirds "don't work" in ET. The key to having them "work" is not to impose an arbitrary value for the F3-A3 M3 but to allow the set of the four contiguous thirds from A3-A4 to discover the right value. When that is done, everything else will fit into place normally and as expected.

Now, with well temperaments, the rules of the game are somewhat different. A well temperament can still be a well temperament and have all of the described and defined properties even if the results vary from one piano to another.When I do the EBVT (any version) aurally, I usually do not go to the extent of measuring the initial 6 beats per second against a metronome although I sometimes do. I just estimate it; it sounds like what I want to hear and go with it. So, there definitely can be some slight variation in it.

However, when I use the programmed tuning/direct interval approach, I always set A3 at 0.0 read on the 4th partial. When I set F3 at 1.0, the F3-A3 M3 inevitably sounds like 6 beats per second to me. The Steinway tuning was measured with a metronome and so was Grandpianoman's. The only time I can recall that 1.0 did not work for F3 was recently when I tuned Andy's Lester spinet.

In that case, there is a long bass bridge with six tri-chord wound strings extending up to F#3. It is a very short scaled piano. The lowest plain wire tenor strings were presumed to have very high inharmonicity while the highest wound string obviously had much lower inharmonicity. In that case, F3 at 1.0 sounded like 7 beats per second to me.

I used a metronome and settled on F3 at 0.0. Normally, one would expect both F3 and A3 as read on the 4th partial and set at 0.0 to beat at approximately 7 beats per second. In this case, the low inharmonicity string had to be tuned sharper than usual to produce 6 beats per second with the high inharmonicity string.

Now, I don't know the higher math and couldn't at this time begin to dabble in it. If I were to set F3-A3 as slow as 5 beats per second, I could still use the EBVT sequence but it would alter the rest of the temperament enough to be more like an early 19th Century temperament. Indeed, I use the very same sequence to tune an representative 18th Century style temperament but start with the initial four RBI's at 4 beats per second.

When I do that, I can leave the F3-A#3 4th beatless. The rest follows exactly the same as the original EBVT. Changing the initial F3-A3 M3 by just 2 beats per second creates an entirely different WT which Owen Jorgensen said was a Young-Werkmeister composite. I can see what he meant by that because it has 6 beatless 5ths like Werkmeister but the initial M3's and M6 are slightly faster like Young's.

I would tune this 18th Century style temperament's octaves the same way I would the EBVT's. That is by equalizing octave and 5th combinations. Therefore, a calculated ETD program would not work well. Some octaves would be too wide and others too narrow. However, a meantone temperament will work well because all of the tempered 5ths are tempered equally.

Accordeur,

It is fine to use the ETD program for tuning the EBVT III. However, you will want to aurally verify the span for F4 to F5. After tuning according to the program, play the 5ths beginning at A#3-F4 to A#4-F5. If there are any that sound too intolerably narrow, you will want to widen the 5th. This may mean simply sharpening the upper note but if the 4th below the note being tuned also sounds quite fast, you may want to flatten that note rather than sharpening the note being tuned. Sometimes you may want to do both. Often, it all sounds acceptable except for the G4-D5 5th. Inevitably, you will want to sharpen D5 by 1 cent.

Using the calculated ETD program for tuning the EBVT III can produce acceptable/satisfactory results but the they are never quite as perfect as with aural tuning or tuning the octaves by direct interval. I am working on a long article that explains all of this in exact detail. It will be presented as a handout to the attendees at my classes in Michigan and at the upcoming convention. The article will also appear on my website when it is finished and will be submitted to the Journal for publication.

Bill, I think the key is that you feel them sounding like 6 bps, but I suspect they might be a tiny bit narrower. Let's settle on 5.5 and be happy all, shall we? Just joking

Anyways, when I tune small pianos in EBVT III, my "beat rate framework" is indeed maybe 1/2 bps slower - somewhere between 5 and 6 bps. Otherwise I get a really strange place for G3, which leads me to having a hard time placing D4, and getting a twangy G3-C4 fourth.

I have to admit that I haven't ever done the math on this one, this is just empirical research.

I will start making the sound files from yesterday's recording in an hour or so.

Kees, I have on paper, the 2 EBVT III tunings Bill did for me...This later one has much more of a stretch than the first one. These are exact figures for all 88 notes. Of course this was designed for my piano. Would these figures help you in any way?

Bill, if the contiguous third method for tuning ET produces F3A3 at 6 bps on a small piano, surely the F3A3 in EBVT must be beating slower, agreed? I programmed it in as a factor 6/7 slower, to agree with your idealized numbers.

Does this sound reasonable? Aurally these numbers are close to what I end up with, although I have to admit it has to be a ridiculously short and quite severely "musically challenged" piano before I go below 5.5 bps. As of today's date, I've done that exactly once... It salvaged my temperament, though, so it was def worth it.

Does this sound reasonable? Aurally these numbers are close to what I end up with, although I have to admit it has to be a ridiculously short and quite severely "musically challenged" piano before I go below 5.5 bps. As of today's date, I've done that exactly once... It salvaged my temperament, though, so it was def worth it.

If you do it then it's reasonable. I have the strange task of recovering a mathematical specification of a temperament from aural tuning instructions, which is the inverse of the usual problem.

Here my questions about the aural tuning instructions for EBVT3,attempting to get everything precise and independent of taste and commonsense. The goal is to make the instructions precise enough to implementin an ETD. The quoted material are Bill Bremmer's instructions.

I'll post questions about the bass later on.

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Just as with the ET via Marpurg, there is only one estimate which youhave to make, the F3-A3 M3. However, rather than estimating it only, youmay actually want to count these beats although I never do. Because Ihave so many years experience, I simply know what 6 beats per secondsounds like. It would not really matter if that interval is exactly 6.0or not but if it is 7.0, it will not work and if it is 5.0, it would endup being another style temperament. So, you do have some margin forerror and your instincts alone should tell you when you are within thatmargin.

The sound is just slightly slower, sweeter and more appealing than an ETF3-A3 M3. That is what I mainly go by. The ET F3-A3 M3 has this kind of"off cadence" sound to it. Yet, when the same interval has 6 beats init, it seems to fall into line with my perception of a one second timeinterval. As a musician, you probably have a goo sense too of what atempo of 60 is. You can use a metronome set at 120 and see if there seemto be 3 beats per tick.

Once you are satisfied with that F3-A3 M3, you need not count any morebeats nor estimate any more intervals.

Here I take the F3A3 beatrate to be 6/7 of the beatrate resulting froman ET. On concert grands it is close to 6, but on tiny little spinitesit may become 5.

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Next, you tune C4 from F3 as a pure 5th which is very easily done. Ofcourse you may use a test interval to prove that it is really pure andstill not slightly tempered or slightly wide. That is suggested.

In the instructions for the temperament scheme we should state what kindof "pure fifths". I assume they are to be 3:2.

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Next, you tune F4 from F3 as a pure sounding octave. When that octavesounds pure and the C4-F4 4th is also perfectly pure, you have the rightsound and it will be a 4:2 octave. You can prove that with a testinterval if you like.

Now, you will listen to the F3-A3 M3 and just "copy and paste" thatsound to C4-E4. Don't count beats any more. Tune the C4-E4 M3 so thatits beats sound exactly the same as those of the F3-A3 M3. The resultwill be that the A3-E4 5th will sound tempered and it will definitely bemore tempered than it would be in ET. If you are tuning the EBVT III,you will improve that 5th at the second to last step but for the EBVT,the tempered sound you hear is correct. Don't worry that it doesn'tsound as "clean" as it does in ET, it is not supposed to and if it does,it isn't correct.

Once you have tuned E4, you will tune G3 to it as a M6 and once again,you will "copy and paste" that sound as precisely as you can. Don'tcount, listen and copy that sound. The result will be that the G3-C4 4thwill be more tempered than it would be in ET. That is correct, so don'tworry that it does not sound the same as ET, it is not supposed to. Youcan refer to the graphs for the expected theoretical beat rate if youwant, to assure yourself that a slightly wider than ET 4th is correctand expected.

Now, listen again to the initial F3-A3 M3. When you tune B3 from G3,again, "copy and paste" that same sound. Don't count beats, you weredone with that when you tuned F3-A3.

Now, listen to these 4 Rapidly Beating Intervals (RBI). They should allhave the same sound and character. If any one is faster or slower thanthe others, make any needed adjustments until they all sound as exactlyalike as possible. You now have the foundation for the EBVT, just as youwould have a foundation for ET with the CM3s. The rest is much easier,just as it is in ET after you are satisfied with the CM3s.

For D4, I suggest that you first temporarily tune D4 as a pure 4th fromA3 (no need to test because it is temporary). Now listen to theresultant G3-D4 5th and notice the strong beat. Sharpen D4 until boththe G3-D4 5th and the A3-D4 4th beat exactly the same. That is veryeasily done. Compare each interval and adjust D4 until both intervalshave the exact same tempered sound. Don't count beats, listen for thesame quality of sound only. Neither will be the same as in ET but reallynot that much different.

For A#3, again temporarily tune it as a pure 4th from F3 (no need toprove it). Go back to the G3-C4 4th, listen to that tempered sound andthen sharpen A#3 until you have "copied and pasted" that exact sametempered sound to the F3-A#3 4th. Again, it will not be the same as ETbut not very different either.

Now, listen to the resultant A#3-D4 M3. It will be just slightly slowerthan you would expect for ET but not very different. A nice pleasantlybeating M3. Tune C#4 from A3 also as a M3 by again, "copying andpasting" that very same tempered sound. Don't count beats. Create thevery same tempered sound. It will be approximately the same as you wouldexpect in ET but be very sure that both the A#3-D4 M3 and the A3-C#4 M3sound exactly alike as possible. In ET, they would not sound exactlyalike but for the EBVT, exactly the same is correct.

Now tune F#3 from C#4 as a pure 5th which is quite easily done but youshould prove that the 5th is pure and not slightly wide. The resultantF#3-A#3 M3 will sound quite fast compared to the same in ET but don'tworry, it is correct. It is supposed to sound that way. The F#3-A#3 andA#3-D4 CM3s would sound wildly inverted compared to ET but they aresupposed to be wildly inverted, so you need not use those testintervals, they are not valid for the EBVT.

Now tune G#3 from C#4 as a pure 4th which is also easily done but youshould prove that the 4th is really pure and not slightly wide ornarrow. The resultant G#3-C4 M3 will sound much faster than it does inET but again, it is supposed to be that way. The G#3-C4 and C4-E4 CM3swould again sound wildly inverted so you need not use those testsbecause they are not valid for the EBVT.

The last step for the EBVT is to tune D#4. I suggest you temporarilytune D#4 from A#3 as a pure 4th (no need to test). Now compare theA#3-D#4 4th to the G#3-D#4 5th. The 5th should sound slightlytempered. Sharpen D#4 very slightly until both the G#3-D#4 5th and theA#3-D#4 4th have the same exact tempered sound. This time, that temperedsound will be very nearly pure, less tempering than you would expectfrom ET.

The above is all crystal clear.

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You now have the original EBVT. It is an excellent temperament whichcontains 5 pure intervals! F3-C4, F#3-C#4, G#3-C#4, B3-E4 andC4-F4. However, there was some feedback that it is a bit too harsh inthe remote keys. It is still excellent for playing all types of musicbut if Romantic era music sounds a bit too strained for you, simplyfollow the next two steps and that will convert the EBVT to the EBVTIII.

Play the A3-E4 tempered 5th and compare it to the B3-E4 pure4th. Sharpen E4 slightly until both the A3-E4 5th and the B3-E4 4th haveexactly the same tempered sound. Don't count beats, just create the sametempered sound in both intervals. Both will be slightly more temperedthan they are in ET but the difference is small. That creates the EBVTII but I never stop with that alone anymore because the key of F# isstill too harsh and it should be changed as well. Therefore, the EBVT IIwas a transitional step for me in trying to find a way to make the EBVTa little milder. It only goes half way in doing so. Therefore it is nolonger used or suggested.

Finally, Listen to the F#3-C#4 pure 5th and compare it to the F#3-B44th. Sharpen F#3 slightly so that the F#3-C#4 5th and the F#3-B4 4th nowhave the exact same tempered sound. Don't count beats just create thevery same tempered sound in both intervals. You now have the EBVT III.

How should the 4th and 5th be tuned to have "exactly the same temperedsound"? Equal beating?

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OK, here it goes for exactly how I tune the octaves from the F3-F4temperament of the EBVT or EBVT III by ear. I will save these drafts anduse them as the basis for posting them formally on my website and thenfor a future PTG Journal article on the subject.

What I am about to say may look at little unusual to you but it isexactly what I do and how I create the most beautiful sound I can fromthe piano. I have done it this way for 20 years. In 1998 at theTemperament Festival at the PTG Convention held in Providence, RI, theEBVT made it to the final round and was compared to a piano tuned in ETby Virgil Smith. The final vote from an audience made up entirely ofpiano technicians favored the EBVT with Tempered Octaves 4:1. The EBVTwith tempered octaves was again presented at the PTG conventions in 2006and 2008. I consider the tempered octaves to be just as important partof the whole sound of the piano as the temperament itself.

Tuning from E3 to C3.

After completing the F3 to F4 temperament octave, begin tuning the restof the low tenor starting with E3. First tune a reasonable soundingoctave from E4, then compare E3 with the 4th and 5th above it and adjustE3 so that the octave still sounds reasonable but the 4th and 5th beatexactly the same or as nearly to that as possible. You can cause the 5thto be slightly less tempered sounding than the 4th but not at theexpense of creating an obvious beat in the octave. The important thingis to have all three, octave, 4th and 5th sound reasonable.

Two issues:(1) What do you want ideally? Have the 5th and 4th beat proportionallyaccording to the rations of ET, or beat exactly equal?

(2) What is a "reasonable octave" precisely? I checked in my softwarethat the 1:2 or 4:2 octave does not beat faster than either the 4th or5th. In practice the octave beat rate is always much slower, so thischeck is superfluous (for a computer which makes no errors).

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You do not need to check any RBIs. The M3s, M6s and minor thirds (m3)will all sound uneven if played chromatically. No RBI test that would benecessary in ET would ever be valid when tuning the EBVT. Just as a 17thor 18th Century tuner who would not have known those tests, you do notneed to use them at all, just skip that entirely. Whatever happens tothe RBIs does not matter.

Continue likewise for D#3 and D3.

So D#3 and D3 are tuned with the same method as E3.

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At C#3, the F#3-C#4 5th is pure in thetemperament octave, so when you tune C#3 to C#4 as a reasonable soundingoctave, you should find that the C#3-F#3 4th also sounds pure. You mayflatten C#3 just enough to slightly improve the C#3-G#3 5th but not atthe expense of creating an obvious beat in the octave nor the 4th. The4th need not remain perfectly pure but it should also not have anobviously tempered sound.

This is unclear. Should the 4th and 5th above C#3 have the same relationas for D3? If so, how does tuning C#3 differ from tuning D3? In practicethe octave is never a problem here, but I need to know if the 4th aboveC#3 should be purer than the 5th above it, or the same, or in the usualET beat proportion.

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Tuning C3 is similar. The F3-C4 5th is pure in the temperamentoctave. Therefore when you tune C3 from C4 as a reasonable soundingoctave, you should find that that the C3-F3 4th is pure. Similar totuning C#3, You can slightly flatten C3 to slightly improve the C3-G35th but again, not at the expense of creating an obvious beat in theoctave nor the 4th. The 4th does not need to remain perfectly pure butit should also not have an obviously tempered sound.

I understand whatever method is used for C#3 should be u8seed for C3.

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The region just above the temperament octave is often called the"Killer" octave because of its difficulty in making compromises both intuning and voicing. I think of the "Killer" octave as being F4 to F5 butsome may think of it as an octave and a half or as much as two fulloctaves.

Begin with F#4 similar to the way you tuned down from the temperamentoctave. When you tune F#4 from F#3 first as a reasonable soundingoctave, the C#4-F#4 4th should sound pure but the B3-F#4 5th will soundtempered. You may sharpen F#4 slightly so that the B3-F#4 5th isslightly improved but not at the expense of creating an obviously wideoctave or an obviously tempered 4th. The 4th need not remain perfectlypure but it should also not be obviously tempered.

So what is the desired relation between the beat speed of C#4-F#4 andB3-F#4? Equal? 4th slower? 5th slower?

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When you tune G4 from G3 as a reasonable sounding octave, both the 4thand 5th below it will sound tempered. Now, you will definitely want toimprove the 5th as much as possible by sharpening G4 slightly. Check tosee that the octave still sounds reasonable but it can be allowed tohave a slight beat in it. I do not specify an exact size of octave butif you do this properly, the size of the octave would be in the 6:3range. More about that later. Naturally, the speed of the 4th will beactive and you don't want it to be excessive but it is not nearly asimportant as the sound of the 5th. So, favor slightly the sound of the5th over both the 4th and the octave. The 5th will still be tempered butwhat you want is a tempered sound that is barely perceptible. If youthen play the C3-G4 octave and 5th, that will sound quite good, close topure if not virtually pure.

So what is the desired relation between the beat speeds of the 5th and4th below G4? Not the same as we have to "favor slightly the sound of the5th over both the 4th and the octave". The octave here is always beating much slower at the 2:1and 4:2 level and seems to play no role. In my implementation I made the5th beat 2/3 as fast as the 4th.

When you tune A4, do that similarly to the way you tune G4. The resultwill be a slightly improved 5th below it, a slightly faster 4th below itand the octave will inevitably be a wider 6:3 type if you care to testit but you need not do that. If you followed the instructions literallyfrom the beginning and started with an 6:3 A3-A4 octave, A4 will alreadybe where it needs to be.

So whatever the precise method for G4 is is what we use for A4.

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Tuning A#4 is also easy. Tuning a reasonable sounding octave from A#3will produce a 4th and 5th below it which also sound reasonable. You mayvery slightly sharpen A#4 so that the D#3-A#4 5th sounds virtually purebut the F4-A#4 4th is only slightly tempered.

What beat ratio you want here for the D#3-A#4 5th and the F4-A#4 4th?

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Tuning B4 is similar to tuning G4 and A4. Favor the 5th below itslightly over the 4th. The E3-B4 octave and 5th will sound virtuallypure.

So whatever the precise method for G4 is is what we use for A4.

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When you tune C5 from C4 as a reasonable sounding octave, the F4-C5 5thwill be pure. Leave it that way. The C3-C5 double octave will also soundvirtually or perfectly pure. Play a C Major arpeggio from C3 to C5 ,hold it with the pedal and you will hear a beautiful chord that largelyor entirely suppresses all of the RBIs within it. It will sound asthough you have tuned that chord in 1/4 or 1/5 comma meantone but youhave not. It only sounds like you have. The equal beating RBIs cancelthemselves out form your perception.

Tuning C#5 is easy. When you tune a reasonable sounding octave from C#4,the F#4-C#4 5th will sound virtually pure. Since you may have sharpenedF#4 slightly, simply tune the F#4-C#5 5th as pure and check to see thatthe C#4-C#5 octave still sounds reasonable.

Clear.

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Tuning D5 is always the most difficult compromise to make. The G3-D4 5thin the temperament octave is tempered more than twice as much as in ET(more than 4 cents narrow) in the EBVT and in the EBVT III it is stilltempered by close to 3 cents narrow. You have already sharpened G4slightly to improve the C4-G4 5th. So, now to create a G4-D5 5th thatdoes not offend the ear, you must compromise the D4-D5 octave. Whentuning electronically, I have found that it most often means adding onefull cent over what the default stretch would provide. That is not anexcessive amount.

It will cause the D4-D5 octave to have about one beat per second init. Played as an isolated octave, an ET tuner may find thatobjectionable but remember, you are not tuning ET! You are not takingthe tuning exam! What is most important is that the G4-D5 5th not soundtoo narrow and have an obvious beat in it. As a practical matter, youcould consider that the octave is one beat per second wide and the 5this one beat per second narrow. What happens to the 4th below D5 does notmatter. This is high enough in the scale that the sound of the 4th isbeginning to fade from perception. The D3-D5 double octave may also beslightly wide.

The way I understand it, we should tune and equal beating D4-D5 4:2 octave andG4-D5 3:2 fifth here.

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The audible width of the octave and double octave are not nearly asimportant as creating a 5th that does not offend the ear. Virtuallyanyone but especially a string player would not want to hear an obviousbeat in the G4-D5 5th but they wouldn't care at all that the octave anddouble octave are slightly wide. You can also consider that the width ofthe D4-D5 octave is about the same as would be created when tuning theET with pure 5ths idea. So, some ET tuners would, in fact have about thesame width in this octave at this point, so it is not extreme.

By now you must have seen that each of these octaves tuned so far is ofa different size (width). That is, in fact, correct and it is in fact,intended. No ETD program, at least as far as I know, can do this. If youwant to use an ETD calculated program, you can do so but what you willwant to do is examine each of these combinations of octaves, 4ths and5ths. You will want to sharpen some notes slightly and flatten othersslightly. When you find that a note needs to be sharpened or flattened,either do that by ear, find the pitch with the ETD and enter the newvalue in the program or estimate how much flatter or sharper the noteshould be with the ETD, tune the note to that pitch and check itaurally. If it now seems correct, enter that value in the program ormake another estimate until you are satisfied, then enter it in theprogram. The amount of change to each note would be in the range of 0.5cents to 1.5 cents. I cannot imagine needing to change any note any morethan 2.0 at the very most.

Continue with D#5 which will be easy. Tune a reasonable sounding octaveand check with the 5th below it. If the 5th sounds too tempered, sharpenD#5 slightly so that it is improved but the octave still soundsreasonable. At this point in the scale, the 4th below the note beingtuned is completely irrelevant. Only check to make sure the doubleoctave is not overly wide. The octave and 5th should sound quite good.

Tune E5 in the same manner as you have been. Pay special attention tothe A4-E5 5th and make sure it does not beat objectionably.

The way I understand it, we should D#5 and E5 with the same method asD5.